Simplify each expression. See Example 1. (a⁸)(a⁵)(a)

Verified step by step guidance

1. Recall the product rule for exponents: when multiplying expressions with the same base, you add the exponents. That is, \(a^m \cdot a^n = a^{m+n}\).
2. Identify the exponents in the expression \((a^8)(a^5)(a)\). Note that \(a\) can be written as \(a^1\) since any variable without an exponent has an exponent of 1.
3. Apply the product rule by adding the exponents: \$8 + 5 + 1$.
4. Write the simplified expression as \(a^{8+5+1}\).
5. Combine the exponents to express the final simplified form as \(a^{14}\) (do not calculate the sum explicitly if you want to stop before the final value).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers of the same base. Specifically, when multiplying terms with the same base, you add their exponents. For example, a^m * a^n = a^(m+n). This rule is essential for simplifying expressions like (a^8)(a^5)(a).

Base Consistency

When applying exponent rules, the base must be the same for the rules to apply directly. In the expression (a^8)(a^5)(a), all terms have the base 'a', allowing the exponents to be combined. Recognizing consistent bases is crucial before simplifying.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their simplest form by applying algebraic rules. In this case, combining like terms with exponents helps write the expression more compactly and clearly, which is a fundamental skill in algebra.